The process that you have been through will automatically amend Motability’s records and the insurers. I phoned both just to be sure, but they said don’t worry it happens automatically. After 5 days the online account still showed the private plate (which I had removed, but the process is the same) so I called Motability and the lady I spoke to sent a chaser to the relevant department and it changed later that day.
I was also concerned about being stopped by the Police, but apparently their systems communicate with the DVLA and will know that a change is in progress. Besides, normal police cars tend not to have ANPR, it’s just the cars they use for motorways / armed response etc, so your chances of it ‘pinging’ are very remote and even then they would only stop you if they had nothing to do, which is almost never these days!